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If the result of $a . b$ ist an odd number, prove both $a$ and $b$ are odd numbers. Thank you in advance!

user577215664
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nguyen minh
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    Have you ever seen a proof by contraposition or contradiction? Can you show that if at least one of $a$ and $b$ are even that their product is even? Do you understand why this is equivalent to your original statement you are being asked to prove? – JMoravitz Sep 16 '20 at 14:31

2 Answers2

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Let $k=a\cdot b$. If either $a$ or $b$ or both are even, i.e. they are divisible by $2$ then $k$ would also be divisible by $2$. But given that $k$ is an odd number, it cannot be divisible by $2$. Hence either $a$ or $b$ or both cannot be even. Both must be odd. $\square$

Natasha
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If $ab$ is even, then $2$ divides $ab$ so either $2$ divides $a$ or $2$ divides $b$ (this is a fundamental property of prime numbers). So if the product is even, at least one, and possibly both, of $a$ and $b$ is even. The remaining case has both $a$ and $b$ odd. So, $a=2m+1$ and $b=2n+1,$ whence $ab = 4mn +(2m+2n)+1,$ which is plainly odd.

Chris Leary
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